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perceived by mobile users for new services in cellular networks
Miodrag Jovanovic
To cite this version:
Miodrag Jovanovic. Evaluation and optimization of the quality of service perceived by mobile users for
new services in cellular networks. Networking and Internet Architecture [cs.NI]. Télécom ParisTech,
2015. English. �NNT : 2015ENST0052�. �tel-01238450�
T H È S E
2015-ENST-0052
EDITE - ED 130
Doctorat ParisTech T H ` E S E
pour obtenir le grade de docteur d´ elivr´ e par
TELECOM ParisTech
Sp´ ecialit´ e “Informatique et R´ eseau”
pr´ esent´ ee et soutenue publiquement par
Miodrag JOVANOVIC
le 11 Septembre 2015
Titre
Evaluation et optimisation de la qualit´ e de service per¸ cue par les utilisateurs mobiles pour les nouveaux services dans les
r´ eseaux cellulaires
Directeur de th` ese: Bartek BLASZCZYSZYN Co-encadrement de la th` ese: Mohamed KARRAY
Jury
M. Laurent DECREUSEFOND, Professeur, T´ el´ ecom ParisTech Pr´ esident du jury M. Sem BORST, Professeur, Technische Universiteit Eindhoven et Bell Labs Rapporteur M. Martin HAENGGI, Professeur, University of Notre Dame Rapporteur M. M´ erouane DEBBAH, Professeur, Sup´ elec et Huawei Examinateur
M. James ROBERTS, Professeur, IRT-SystemX Examinateur
M. Bartek BLASZCZYSZYN, Professeur, Ecole normale sup´ erieure et INRIA Directeur de th` ese TELECOM ParisTech
´ ecole de l’Institut Mines-T´ el´ ecom - membre de ParisTech
46 rue Barrault 75013 Paris - (+33) 1 45 81 77 77 - www.telecom-paristech.fr
To my family and friends
3 RESUME: L’objectif de cette th` ese est de d´ evelopper des outils et des m´ ethodes pour l’´ evaluation de la qualit´ e de service (Quality of Service - QoS) per¸ cue par les utilisateurs, en fonction de la demande de trafic, dans les r´ eseaux cellulaire sans fil moderne. Ce probl` eme complexe, directement li´ ee au dimensionnement du r´ eseau, implique la mod´ elisation des processus dynamiques ` a plusieurs ´ echelles de temps, qui en raison de leurs nature al´ eatoire se prˆ etent ` a la formalisation probabiliste.
Tout d’abord, sur la base de la th´ eorie de l’information, nous capturons les performances d’un seul lien entre une station de base et un utilisateur dans un r´ eseau cellulaire avec des canaux orthogonaux et la technologie MIMO. Nous prouvons et utilisons certaines bornes inf´ erieures de la capacit´ e ergodique en vue de la th´ eorie de l’information d’un tel lien, qui prend aussi en compte la variabilit´ e du canal rapide caus´ ee par la propagation des trajets multiples. Ces bornes donnent une base solide pour l’´ evaluation plus profonde de la qualit´ e de service per¸ cue par les utilisateurs.
Ensuite, on consid` ere plusieurs utilisateurs (´ eventuellement mobiles), arrivant dans le r´ eseau et de- mandant un service. Nous consid´ erons des services (´ elastiques) ` a d´ ebit variable dans lesquels les trans- missions de certaines quantit´ es de donn´ ees sont r´ ealis´ ees d’une mani` ere ”best-effort”, ou services ` a d´ ebit constant, dans lesquels une certaine vitesse de transmission doit ˆ etre maintenue pendant les p´ eriodes demand´ ees. Sur la base de la th´ eorie des files d’attente, on capture cette demande du trafic et processus de service ` a l’aide des mod` eles appropri´ es (multi-classes) de partage du processeur (processor-sharing PS) ou mod` ele de perte. Dans cette th` ese, nous adaptons les mod` eles PS existants et d´ eveloppons un nouveau mod` ele de perte pour le trafic streaming de transmission sans fil, o` u les bornes th´ eoriques (au regard de la th´ eorie de l’information) mentionn´ ees ci-dessus de la capacit´ e des liens simples d´ ecrivent les taux de service instantan´ es des utilisateurs. Les mod` eles multi-classes sont utilis´ es pour capturer l’h´ et´ erog´ en´ eit´ e spatiale des canaux utilisateur. Ceux-ci d´ ependent de l’emplacement g´ eographique de l’utilisateur et du ph´ enom` ene de ”shadowing” de propagation.
Enfin, au-dessus des processus de file d’attente th´ eoriques, il faut tenir compte d’un r´ eseau multicel- lulaire, dont les stations de base ne sont pas n´ ecessairement r´ eguli` erement plac´ ees, et dont la g´ eom´ etrie est en outre perturb´ ee par la ph´ enom` ene de shadowing. Nous abordons cet aspect al´ eatoire en utilisant des mod` eles de g´ eom´ etrie stochastique, notamment processus de Poisson ponctuels et le formalisme de Palm appliqu´ e ` a la cellule typique du r´ eseau. En appliquant l’approche triple mentionn´ ee ci-dessus, cens´ ee ` a repr´ esenter tous les m´ ecanismes cruciaux et les param` etres de l’ing´ enierie des r´ eseaux cellu- laires (tels que LTE - Long Term Evolution), nous ´ etablissons des relations macroscopiques entre la demande de trafic et les m´ etriques de la qualit´ e de service per¸ cue par les utilisateurs pour certains ser- vices ` a d´ ebit binaire ´ elastiques et constants. Ces relations sont obtenues principalement d’une mani` ere semi-analytique, c’est-` a-dire qu’elles concernent des simulations statiques d’un processus ponctuel de Poisson (mod´ elisation des emplacements des stations de base). Ceci afin d’´ evaluer ses caract´ eristiques qui ne se prˆ etent pas aux expressions analytiques.
Plus pr´ ecis´ ement, en ce qui concerne le trafic de donn´ ees (le service de d´ ebit binaire ´ elastique), nous capturons l’interf´ erence inter-cellule, rendant les mod` eles des files d’attente PS de cellules individuelles d´ ependantes, via un syst` eme d’´ equations de charge des cellules. Ces ´ equations permettent de d´ eterminer le d´ ebit moyen par utilisateur, le nombre moyen d’utilisateurs et la charge moyenne de la cellule dans un grand r´ eseau, en fonction de la demande du trafic. La distribution spatiale de ces m´ etriques de QoS dans le r´ eseau est ´ egalement ´ etudi´ ee. Nous validons notre approche en comparant les r´ esultats obtenus avec ceux mesur´ es ` a partir de traces du r´ eseau r´ eel. Nous observons une concordance remarquable entre les pr´ edictions du mod` ele et les donn´ ees statistiques recueillies dans plusieurs sc´ enarios de d´ eploiement.
En ce qui concerne les services de d´ ebit binaire constants, nous proposons un nouveau mod` ele stochas- tique pour ´ evaluer la fr´ equence et le nombre d’interruptions lors de streaming en temps r´ eel en fonction des conditions radio utilisateur. Nous l’utilisons pour ´ etudier les m´ etriques de la qualit´ e de service en fonction des conditions radio utilisateur dans les r´ eseaux LTE.
Tous les r´ esultats ´ etablis ici contribuent au d´ eveloppement de m´ ethodes de dimensionnement de r´ eseau et sont actuellement utilis´ es dans les outils internes d’Orange pour les calculs de capacit´ e du r´ eseau.
MOTS-CLEFS: QoS; LTE; d´ ebit; charge; th´ eorie des files d’attente; g´ eom´ etrie stochastique; mesures;
3GPP
ABSTRACT: The goal of this thesis is to develop tools and methods for the evaluation of the QoS (Quality of Service) perceived by users, as a function of the traffic demand, in modern wireless cellular networks. This complex problem, directly related to network dimensioning, involves modeling dynamic processes at several time-scales, which due to their randomness are amenable to probabilistic formalization.
Firstly, on the ground of information theory, we capture the performance of a single link between a base station and a user in the context of a cellular network with orthogonal channels and MIMO technology. We prove and use some lower bounds of the information-theoretic ergodic capacity of such a link, which account also for the fast channel variability caused by multi-path propagation. These bounds give robust basis for further user QoS evaluation.
Next, one considers several (possibly mobile) users, arriving in the network and requesting some service from it. We consider variable (elastic) bit-rate services, in which transmissions of some amounts of data are realized in a best-effort manner, or constant bit-rate services, in which a certain transmission rate needs to be maintained during requested times. On the ground of queuing theory, one captures this traffic demand and service process using appropriate (multi-class) processor sharing (PS) or loss models.
In this thesis, we adapt existing PS models and develop a new loss model for wireless streaming traffic, in which the aforementioned information-theoretic capacities of single links describe the instantaneous user service rates. The multi-class models are used to capture the spatial heterogeneity of user channels, which depends on the user geographic locations and propagation shadowing phenomenon.
Finally, on top of the queueing-theoretic processes, one needs to consider a multi-cellular network, whose base stations are not necessarily regularly placed, and whose geometry is further perturbed by the shadowing phenomenon. We address this randomness aspect by using some models from stochastic geometry, notably Poisson point processes and Palm formalism applied to the typical cell of the network.
Applying the above three-fold approach, supposed to represent all crucial mechanisms and engineer- ing parameters of cellular networks (such as LTE), we establish some macroscopic relations between the traffic demand and the user QoS metrics for some elastic and constant bit-rate services. These relations are mostly obtained in a semi-analytic way, i.e., they only involve static simulations of a Poisson point process (modeling the locations of base stations) in order to evaluate its characteristics which are not amenable to analytic expressions.
More precisely, regarding the data traffic (the elastic bit-rate service), we capture the inter-cell interference, making the PS queue models of individual cells dependent, via some system of cell-load equations. These equations allow one to determine the mean user throughput, the mean number of users and the mean cell load in a large network, as a function of the traffic demand. The spatial distribution of these QoS metrics in the network is also studied. We validate our approach by comparing the obtained results with those measured from live-network traces. We observe a remarkably good agreement between the model predictions and the statistical data collected in several deployment scenarios.
Regarding constant bit-rate services, we propose a new stochastic model to evaluate the frequency and the number of interruptions during real-time streaming calls in function of user radio conditions.
We use it to study the quality of service metrics in function of user radio conditions in LTE networks.
All established results contribute to the development of network dimensioning methods and are currently used in Orange internal tools for network capacity calculations.
KEY-WORDS: QoS; LTE; throughput; load; queueing theory; stochastic geometry; measures; 3GPP
Contents
1 Introduction 13
1.1 Thesis motivation . . . . 13
1.2 Thesis contribution and structure . . . . 14
2 Link quality 17 2.1 Introduction . . . . 17
2.2 Related work . . . . 18
2.3 OFDM Cellular network with MIMO . . . . 18
2.3.1 Network model . . . . 18
2.3.2 Link capacity given fading . . . . 20
2.3.3 Ergodic capacity . . . . 20
2.4 MMSE . . . . 23
2.4.1 MMSE capacity given fading . . . . 24
2.4.2 MMSE ergodic capacity . . . . 25
2.4.3 MMSE-SIC . . . . 25
2.5 Numerical results for the link capacity . . . . 26
2.5.1 Link layer model calibration . . . . 26
2.5.2 Comparison to simulation . . . . 27
2.5.3 Comparison to measurements . . . . 27
2.5.4 Approximate link quality estimation via simulations . . . . 28
2.6 Link quality observed by a typical user . . . . 30
2.6.1 SINR . . . . 30
2.6.2 Spectral efficiency . . . . 32
Appendices 35 2.A Theoretical results: MIMO flat-fading channel with additive noise . . . . 35
2.1.1 Model . . . . 35
2.1.2 Capacity lower bound . . . . 35
2.1.3 Asymptotic analysis . . . . 39
3 User throughput versus traffic demand — global network performance via a fixed point problem 41 3.1 Introduction . . . . 41
3.2 Related work . . . . 43
3.2.1 Related work regarding the dimensioning problem . . . . 43
3.2.2 Related work regarding QoS evaluation . . . . 44
3.3 Processor-sharing queue model for one cell scenario . . . . 45
3.3.1 No mobility case . . . . 45
5
3.3.2 Infinite mobility . . . . 49
3.4 Multi-cell scenario: symmetric networks . . . . 50
3.4.1 Cell load . . . . 51
3.5 Multi-cell irregular networks scenario . . . . 53
3.5.1 Network model . . . . 53
3.5.2 Generalization of processor sharing model . . . . 53
3.5.3 Global network characteristics . . . . 55
3.5.4 Mean cell . . . . 59
3.6 Heterogeneous networks . . . . 60
3.6.1 Model description . . . . 61
3.6.2 Typical and mean cell in multi-tier network . . . . 62
3.6.3 Full interference model . . . . 64
3.6.4 Weighted interference model . . . . 68
3.7 Numerical results: network dimensioning and QoS estimation . . . . 69
3.7.1 Validation in a dynamic context . . . . 70
3.7.2 Hexagonal LTE network dimensioning . . . . 72
3.7.3 Mean performance estimation of irregular network using Poisson process . 75 3.7.4 Numerical results for heterogeneous networks . . . . 81
3.7.5 Spatial distribution of QoS parameters averaged over many cells in the network . . . . 87
Appendices 93 3.A Proof of Proposition 4 in the Markovian case . . . . 93
4 Quality of service in real-time streaming 97 4.1 Introduction . . . . 97
4.2 Related work . . . . 98
4.3 Streaming in wireless cellular networks . . . . 99
4.3.1 System assumptions . . . . 99
4.3.2 Model description . . . . 100
4.3.3 Model evaluation . . . . 103
4.4 Quality of real-time streaming in LTE . . . . 106
4.4.1 LTE model and traffic specification . . . . 107
4.4.2 Performance evaluation . . . . 109
Appendices 121 4.A A general real-time streaming (RTS) model . . . . 121
4.1.1 Traffic demand . . . . 121
4.1.2 Resource constraints and outage policy . . . . 122
4.1.3 Performance metrics . . . . 122
4.1.4 Mathematical results . . . . 123
5 Conclusion and future work 127
List of Figures
2.1 Jensen (2.8) and asymptotic (2.7) lower bounds for the capacity (peak bit-rate) of the MIMO flat fading channel with additive noise in the downlink of an OFDMA cellular network. Capacity as function of the distance between the user and his
serving base station. . . . 23
2.2 Performance of SISO without fading evaluated using analytic approximation (2.15) and link simulation. . . . . 27
2.3 Analytic relation of the peak bit-rate to SINR compared to 3GPP simulation . . 28
2.4 Analytic relations of the peak bit-rate to SINR compared to measurements; see Section 2.5.3 for the details. . . . . 29
2.5 Simulations versus the analytical expression (right-hand side of (2.19)) for the calibration case . . . . 30
2.6 CDF of the coupling gain (antenna gain minus propagation loss) . . . . 31
2.7 CDF of SINR . . . . 32
2.8 CDF of normalized user throughput . . . . 33
2.9 Comparison of the analytic capacity and the asymptotic formula . . . . 39
3.1 Load versus traffic demand per cell . . . . 71
3.2 Mean user throughput versus traffic demand per cell . . . . 72
3.3 95% quantile of user throughput versus traffic demand . . . . 72
3.4 Mean user throughput as function of the cell radius for different load situations . 73 3.5 Mean user throughput as function of the cell radius for different traffic demand densities (adapted load) . . . . 73
3.6 Cell radius versus traffic demand density for mean user throughput 10
4kbit/s . . 74
3.7 Cell radius versus traffic demand density for different mean user throughputs . . 75
3.8 Local user throughput versus local traffic demand for some zone (selected to sat- isfy a spatial homogeneity of the base stations) of an operational cellular network deployed in a big city in Europe. 9288 different points correspond to the mea- surements made by different sectors of different base stations during 24 different hours of some given day. . . . 76
3.9 Cell load and the stable fraction of the network versus traffic demand per cell in the full interference model. . . . . 77
3.10 Number of users per cell versus traffic demand per cell in the full interference model. 77 3.11 Mean user throughput in the network versus traffic demand per cell in the full interference model. . . . 78
3.12 Load and the stable fraction of the network versus traffic demand in the weighted interference model. Also, load estimated from real field measurements. . . . 78
7
3.13 Number of users versus traffic demand per cell in the weighted interference model.
Also, the same characteristic estimated from the real field measurements. . . . . 79
3.14 Mean user throughput in the network versus traffic demand per cell in the weighted interference model. Also, the same characteristic estimated from the real field measurements. . . . . 79
3.15 Mean user throughput in the network versus traffic demand per area for an urban zone of a big city in Europe. (The density of base stations is 4 times smaller than in the dense urban zone considered in Figure 3.14). . . . 80
3.16 Ripley’s L-function calculated for the considered dense urban and urban network zones. (L function is the square root of the sample-based estimator of the expected number of neigbours of the typical point within a given distance, normalized by the mean number of points in the disk of the same radius. Slinvyak’s theorem allows to calculate the theoretical value of this function for a homogeneous Pois- son process, which is L(r) = r.) In fact, in large cities spatial, homogeneous “Poissonianity” of base-station locations is often satisfied “per zone” (city center, residential zone, suburbs, etc.). Moreover, log-normal shadowing further justifies the Poisson assumption, cf. [20, 29] . . . . 81
3.17 Accuracy of the homogeneous approximation of the mean cell . . . . 82
3.18 Cell load versus traffic demand per cell in the full interference model. . . . 83
3.19 Number of users per cell versus traffic demand per cell in the full interference model. 83 3.20 Mean user throughput in the network versus traffic demand per cell in the full interference model. . . . 84
3.21 Cell load versus traffic demand per cell in the weighted interference model. . . . 85
3.22 Number of users per cell versus traffic demand per cell in the weighted interference model. . . . 85
3.23 Mean user throughput in the network versus traffic demand per cell in the weighted interference model. . . . 86
3.24 CDF of BS powers in the operational network in the downtown of a big city (blue) and normal distribution approximation (red). . . . 88
3.25 CDF of cell load for the downtown of a big city obtained either from the vari- able power model, from real-field measurements, or from the model where the transmitted powers are assumed constant. . . . 89
3.26 CDF of the mean users number for the downtown of a big city. . . . 90
3.27 CDF of the throughput for the downtown of a big city. . . . 90
3.28 CDF of cell load for a mid-size city obtained either from the variable power model, from real-field measurements, or from the model where the transmitted powers are assumed constant. . . . . 90
3.29 CDF of the mean number of users for the mid-size city. . . . . 91
3.30 CDF of the throughput for the mid-size city. . . . . 91
4.1 Cumulative distribution function of the SINR . . . . 108
4.2 Fraction of time in outage; traffic 900 Erlang/km
2. . . . 110
4.3 Fraction of time in outage; traffic 600 Erlang/km
2. . . . 111
4.4 Number of outage incidents; traffic 900 Erlang/km
2. . . . 112
4.5 Number of outage incidents; traffic 600 Erlang/km
2. . . . 113
4.6 Deep outage versus outage time . . . . 114
4.7 Mean total throughput . . . . 115
4.8 Deterministic vs Poisson arrivals; fraction of time in outage, 900 Erlang/km
2. . 116
4.9 Deterministic vs Poisson arrivals; fraction of time in outage, 600 Erlang/km
2. . 117
LIST OF FIGURES 9
4.10 Deterministic vs Poisson arrivals; number of outage incidents, 900 Erlang/km
2. 118
4.11 Deterministic vs Poisson arrivals; number of outage incidents, 600 Erlang/km
2. 119
List of Tables
2.1 Results of the linear fittings. . . . 29 2.2 Cell spectral efficiency: Comparison of the 3GPP simulations and the analytic
results. . . . 33 3.1 Mean and standard deviation of spatial distribution of QoS metrics for the down-
town of a big city . . . . 88 3.2 Mean and standard deviation of spatial distribution of QoS metrics for the mid-size
city . . . . 89
11
Chapter 1
Introduction
1.1 Thesis motivation
There is a need for simple, yet realistic methods for the evaluation of the quality of service (QoS) in wireless networks capturing both the spatial distribution of the elements of the network and the temporal dynamics of users and having a limited number of parameters. This can be obtained by decomposing the problem into three layers corresponding to different time-scales, which are addressed on the ground of information theory, queuing theory and stochastic geometry.
Firstly, information theory studies the performance of a single radio link accounting particularly for the signal variations due to multi-path fading. Once the link performance is characterized, resources (power, bandwidth etc.) are allocated to the users while accounting for their mutual interference. This can be modelled by an appropriate service policy on the ground of queuing theory which accounts next for the users’ arrivals, mobility and departures and allows appropriate time averages. Finally, stochastic geometry is used to model network, i.e. base stations spatial pattern and shadowing.
Individual elements of the above puzzle (i.e. information theory, queuing theory and stochas- tic geometry models) are often studied and optimized separately. The main specificity of the methodology proposed in this thesis is a global approach that combines these three elements. In doing so, it is necessary to separate carefully the time scales of different elements of the network dynamics.
We apply an information-theoretic modeling of a link layer between a user and a base station.
In this context we show that the worst additive noise is the white Gaussian one and establish a lower bound for the link capacity. Further the modeling approach consists in representing the configuration of users (positions, call durations or volumes, allocated resources) as a random object (point pattern with associated random variables), which evolves in time. The quality of service perceived by the users may then be expressed as a function of the stationary state of this process and thus will depend only on its distribution parameters. This approach often allows one for an explicit evaluation of the key QoS characteristics and for efficient optimization of the network cost and capacity. Some examples from other research works which prove the pertinence of this approach, can be found in [49], [78], [24], [68], [42], [10], [54]. We use homogeneous spatial Poisson point processes to model base station positions and apply results from stochastic geometry to evaluate QoS.
The performance of wireless cellular networks is often evaluated in terms of parameters such as the spectral efficiency [7] (in particular within 3GPP [3]) or the outage probability [44].
However from the point of view of an operator, it is even more important to calculate the QoS
13
perceived by the users; and in particular to relate this QoS to the key network parameters such as the traffic demand, the cell radius, the transmitted power, etc. This relation is crucial for the network dimensioning; i.e., evaluating the minimal number of base stations (more generally, the required network setup) assuring some QoS (for some given traffic demand). This permits in particular to minimize the network cost. The probabilistic approach described above often allows an explicit evaluation of the key characteristics such as users QoS and for efficient optimization of the network dimensioning.
Classically, the services are classified into two main categories:
• Variable bit-rate (VBR); e.g., mail, ftp. Users aim to transmit some given volume of data at a bit-rate which may be decided by the network
• Constant bit-rate (CBR); e.g., voice, video conferencing. Users require some given (con- stant) bit-rate for some duration. In this case the requested bit-rates may sometimes exceed the available capacity, a situation usually called congestion. CBR services do not tolerate temporary interruptions of their transmissions. Consequently, if congestion occurs, the net- work blocks (i.e., refuses the access to) new calls and/or drops (i.e., interrupts definitely) some other calls during their transmissions.
When we account for calls’ arrivals, mobility and departures, the QoS perceived by the users (in the long run of the network) is different for each of the above traffic classes. For VBR connections, the QoS may be defined in terms of the mean throughput or delay per user [25], [59].
For CBR calls, the main QoS indicators are the blocking and dropping probabilities [10], [42], [59].
The research done in the last few years permitted to build efficient methods to calculate these QoS indicators (see for example [6]). These methods are based respectively on processor sharing for VBR and on multi-class Erlang models for CBR services.
However, new multimedia services are gaining interest in wireless cellular networks, especially streaming services [45]. Streaming connections require some given bit-rate for some duration [68], [75]. Thus congestion may occur (when the bit-rates requested by the users in the network exceed the available capacity). All streaming calls are admitted, but, as a counterpart, they tolerate temporary interruptions of their transmission. We distinguish two sub-classes:
• Streaming-RT (Real-Time): e.g. mobile TV, RTP streaming. When congestion occurs, the corresponding portions of some calls are definitely lost, but the call is not dropped.
• Streaming-NRT (Non-Real-Time): e.g., streaming-video (youtube, dailymotion, on de- mand video) on the web. When congestion occurs, the corresponding portions of calls are delayed.
For the streaming users, the QoS is related to the frequency of the interruption of their calls and the durations of these interruptions. These performance measures depend strongly on the mobility of users, as mobility increases the variability of the radio conditions.
1.2 Thesis contribution and structure
This thesis comprises three technical chapters. In Chapter 2 we focus on the first element of
the analytic approach: single link quality between a user and a base station. We examine
radio links in cellular networks such as LTE (Long Term Evolution) and HSDPA (High-Speed
Downlink Packet Access) and take into account MIMO (Multiple Input Multiple Output) and
OFDM (Orthogonal Frequency Division Multiplex) technologies. The principal result is a lower
1.2. THESIS CONTRIBUTION AND STRUCTURE 15 theoretical bound for a single link quality which is very close to the exact link quality and which is tractable analytically; i.e. calculated in a simple manner.
Once expressions for link quality are developed, we take into account user dynamics using queuing theory. Finally, we apply the results from stochastic geometry to model the spatial configuration of network resources and users. In Chapter 3 and Chapter 4 we apply the above mentioned methodology to evaluate QoS for the different types of services. Namely, in Chapter 3 we develop a method for the user throughput estimation in large cellular networks, regular or irregular, for VBR traffic. In addition, we could estimate the mean number of users and the mean cell load in the network. All the above means account for the disparity of different base stations and traffic randomness over some period of time (one hour, for example). Further, we are able to estimate the spatial CDF (Cumulative Distribution Function) of mean user throughput, mean number of users and cell load over all base stations (averaged in time).
The core of the mathematical modeling in Chapter 3 was to capture the dependence between the traffic demand and the interference in cellular networks with orthogonal channels (in time and/or frequency). We did this using the aforementioned probabilistic tools in order to get analytically tractable and simple relations, but in a manner that reflects the physical behaviour of the system. It turns out that the dependence between the traffic demand and the interference is well captured via a fixed-point problem. Solving this problem, we get all elements to evaluate QoS as function of traffic demand.
In Chapter 4 the evaluation of the QoS for real-time streaming is presented. The number and duration of interruptions are calculated as function of traffic demand and radio conditions, i.e. SINR (Signal to Interference and Noise Ratio). Hexagonal cellular network with orthogonal channels is considered. The stochastic analysis is based on Poisson processes representation of the traffic and Palm formalism related to the typical call. The results can be used to estimate the QoS for this type of traffic, but also for network dimensioning. In fact, based on the spatial distribution of radio conditions we can deduce the QoS at all positions in a cell (area served by one base station) for a given traffic demand. On the other hand the spatial distribution of SINR depends on cell radius. So, determining a constraint on QoS, one can deduce what is the necessary cell radius to satisfy this constraint on QoS. This manipulation can be done for any value of traffic demand, which is the cellular network dimensioning.
Chapter 2 is based on the following publications [61] and [63]. Chapter 3 is constructed from the following articles [62], [15], [57] and [16] and the Chapter 4 is an adapted version of [17].
The results of the thesis are used in Orange tools, such as the operational tool Utrandim,
for the dimensioning of wireless cellular networks. They are also used to study the spectral and
energy efficiencies and the required emitted power in these networks.
Chapter 2
Link quality
2.1 Introduction
In this Chapter we are interested in the link quality analysis. A link is a communication channel between two or more communicating devices. We need it to develop a global analytic approach to the performance evaluation of wireless cellular networks and especially LTE networks.
The link performance in an OFDM (used in LTE) cellular network with MIMO antennas may be studied using two methodologies. Information theory considers the ultimate performance of best possible coding schemes and looks for mathematical formulae to describe this performance.
Real systems, such as 3GPP (3rd Generation Partnership Project) [3], deploy suboptimal coding schemes which are usually evaluated by simulation.
A key link characteristic of OFDM cellular networks is the peak bit-rate at each location defined as the maximal bit-rate a user can get at the considered location from his serving base station. The objective of the present Chapter is to establish some closed form information theoretic bounds for the peak bit-rate in OFDM cellular networks with MIMO and compare them to real system performance predicted by simulation and estimated from field measurements.
We describe a simple model of a MIMO cellular network which permits to obtain an analytical expression of users’ bit-rates, which are feasible from the information theory point of view.
This expression accounts for the variety of MIMO configurations (numbers of transmitting and receiving antennas) and radio conditions (SINR). This expression is compared to practical LTE performance evaluated by 3GPP simulations for different cases including the so-called calibration case [3]. The comparison shows that the analytical expression may be adjusted to the practical performance by a multiplicative coefficient, which depends on the MIMO configuration but not on the SINR. Additionally, we show the progress margin for potential evolution of the technology.
The capacity of the MIMO channel without interference is known. Accounting for the extra- cell interference, Proposition 2 gives a lower bound for the downlink capacity in a multi-cell OFDM network with MIMO antennas. This bound relies on the observation made in Proposi- tion 3 that the worst additive noise for the capacity of the MIMO flat-fading complex-valued channel is the white Gaussian one. In order to make the established lower bound more explicit, we give an asymptotic approximation based on random matrix theory and derive also a further lower bound from Jensen’s inequality.
Finally we build bounds for the peak bit-rate of the MMSE (Minimum Mean Square Error) scheme currently implemented in operational networks as well as its improvement MMSE-SIC (Successive Interference Cancellation).
17
2.2 Related work
Telatar [91] gives the information theoretic capacity of a MIMO channel with fading and AWGN (Additive White Gaussian Noise). Different MIMO configurations are compared for this channel by Foschini and Gans [46]. Blum et al. [22] study the capacity of a MIMO cellular network with flat Rayleigh fading. Clark et al. [32] show that in an OFDM system with a sufficiently large number of sub-carriers, the capacity with respect to Rayleigh fading is approximately normally distributed. Tulino and Verdu [94] apply random matrix theory to analyze this capacity. Random matrix theory is useful to study CDMA as for example in [93], [95] and [43].
The 3GPP [3] evaluates the performance of LTE systems by simulation. Goldsmith and Chua [53] observed that real coding schemes performance may be described by a modification of the famous log
2(1 + SNR) Shannon’s formula. Mogensen et al. [73] have observed that the LTE capacity in the AWGN context is well approximated by this formula with a multiplicative coefficient. These ideas will be extended in the present Chapter to MIMO cellular networks with fading.
Explicit expressions for the capacity of AWGN channels are well known. Interference in wireless cellular networks is not necessarily Gaussian nor white (the term white means that the samples are independent and identically distributed). The explicit expression for capacity in such context is not known. To circumvent this difficulty, a possible idea is to look for a lower bound and check whether it is tight enough to meet a desired precision. Using a result of Shannon [84, Theorem 18], it may be shown that, in a SISO channel, the worst additive noise process with given power is AWGN. This result is extended to a network with relays by Shomorony and Avestimehr [85]. Diggavi and Cover [40] study the worst noise process for an additive channel under covariance constraints and characterize the so-called saddle-point input and noise distributions for the mutual information [40, Theorem II.1]. Girnyk et al. [51] calculate the asymptotic sum-rate of uplink MIMO cellular network.
Note that the fact that the worst additive noise is Gaussian may be derived from [40, The- orem II.1]; but the input and the noise vectors are real-valued there whereas we shall consider complex-valued random vectors. The whiteness of the worst noise process proved in Proposition 3 does not follow immediately from the aforementioned result either.
2.3 OFDM Cellular network with MIMO
2.3.1 Network model
We consider a wireless network composed of several base stations (BS). The power transmitted by each BS is limited to some given maximal value. The network operates the Orthogonal Frequency- Division Multiple Access (OFDMA) linked to OFDM, which we describe now. The frequency spectrum allocated to the considered network is divided into a given number of sub-carriers, which are made available to all base stations. Each BS allocates disjoint subsets of the sub- carriers to its users. Each user is served by a single BS and receives only other-BS interference ; that is the sum of powers emitted by other BS on the sub-carriers allocated to him by his serving BS. We consider multiple input and multiple output (MIMO) antennas. More precisely, BS are equipped with t
Atransmitting antennas whereas users have r
Areceiving antennas and each BS uses all its transmitting antennas to serve a given user.
We assume that the bandwidth of each sub-carrier is smaller than the coherence frequency of the channel, so we can consider that the fading in each sub-carrier is flat [23]. That is, the output of the channel at a given time depends on the input only at the same instant of time.
Indeed, the use of a cyclic prefix in OFDM permits to transform the frequency selective fading
2.3. OFDM CELLULAR NETWORK WITH MIMO 19 channel into a set of parallel flat fading channels [92, §3.4.4]. We don’t make any assumption on the correlation of the fading processes corresponding to different subcarriers for a given user and a given BS. However, the fading processes for different users or base stations are assumed independent.
Time is divided into time-slots of length smaller than the coherence time of the channel, so that, for a given sub-carrier, the fading remains constant during each time-slot and the fading process in different time-slots may be assumed ergodic. Such model for fading generalizes the so-called quasi-static model where the fading process at different time-slots is assumed to be independent and identically distributed. We shall always assume that the receiver knows the fading.
The codeword duration equals the time-slot, which is assumed sufficiently large so that the capacity (peak bit-rate ) within each time-slot may be defined in the asymptotic sense of the information theory.
Users perform single user detection; thus the interference from other BS is added to AWGN.
The statistical properties of the interference are not known a priori since they depend on the coding of other BS.
Consider a user served by a BS indexed by u. For a given sub-carrier and time-slot, we consider the following discrete-time model of the OFDM channel with MIMO [23, Equation (7)]
Y
n= L
−1/2uH
uΨ
n+ Θ
n+ I
n, n ∈ N (2.1) where n is a discrete-time index, Y
n∈ C
rAis the channel output, Ψ
n∈ C
tAis the channel input signal, H
uis a complex matrix of dimension r
A× t
Arepresenting the fading with the serving BS u, I
n∈ C
rAis the interference, Θ
1, Θ
2, . . . are i.i.d. random noises with values in C
rAsuch that each Θ
nis circularly-symmetric Gaussian with covariance matrix E[Θ
nΘ
∗n] = N I
rAwhere Θ
∗ndesignates the transpose complex conjugate of Θ
n, N is a given positive constant and I
rAis the identity matrix of dimension r
A, and L
uis the propagation loss due to distance and shadowing between the user and BS u. The propagation loss L
uis the ratio between the emitted and received powers, hence the factor L
−1/2uin Equation (2.1).The interference equals to
I
n= X
v6=u
L
−1/2vH
vΨ
v,nwhere the sum is over the interfering BS v 6= u, Ψ
v,nis the transmitted signal by the interfering BS v, H
vrepresents fading for BS v, and L
vis the propagation loss due to distance and shadowing between the user and BS v.
We make the following probabilistic assumptions:
(H1) All the channel input signals are centred; i.e. E [Ψ
n] = 0.
(H2) The signals transmitted by different antennas including multiple antennas of the same BS are independent. Let P be the power transmitted by each BS in a given sub-carrier aggregated over all the t
Atransmitters. Assume that this power is equally partitioned between the t
Atransmitting antennas; each one emitting a power P/t
A. This assumption is justified by the last statement in Proposition 3.
(H3) The fading matrices H
vare constant for all channel uses n ∈ N within a given time-slot and sub-carrier. For a given BS v, the fading matrix H
vis resampled across different time- slots, and we assume that it follows a stationary and ergodic sequence of random matrices.
Moreover, these processes are independent across different BS v.
(H4) Users are motionless at the considered information theoretic time-scale; that is L
vare
constant for all BS v and time-slots.
2.3.2 Link capacity given fading
In this section, we focus on a given time-slot and sub-carrier; that is, the expectation E [·] is taken with respect to the distribution of the transmitted signals and noise. By Assumptions (H1-H2), the covariance matrix of the transmitted signals are
E [Ψ
nΨ
∗n] = E
Ψ
v,nΨ
∗v,n= P t
AI
tA(2.2)
and the covariance matrix of the interference equals
E [I
nI
n∗] = E
X
v6=u
L
−1vH
vΨ
v,nΨ
∗v,nH
v∗
= X
v6=u
L
−1vH
vE
Ψ
v,nΨ
∗v,nH
v∗= P t
AX
v6=u
H
vH
v∗L
vNoise and interference are assumed independent, thus the covariance matrix of Σ
n= Θ
n+I
nis
∆=E [Σ
nΣ
∗n] = N I
rA+ P t
AX
v6=u
H
vH
v∗L
v(2.3) The capacity of a channel may be interpreted as a maximal average bitrate sustainable in long communication time. The capacity C is defined in Section 2.6.2.
Proposition 1 The capacity C of the OFDM channel with MIMO (2.1) with power constraint (2.2) is lower bounded by
C ≥ log
2det
I
rA+ P t
AH
uH
u∗L
u∆
−1(2.4) where the noise plus interference covariance matrix ∆ is given by (2.3).
Proof. The mathematical background and proof are defended in Section 2.6.2.
We call the right-hand side of the above equation
1feasible bit-rate for the considered user.
Since our assumptions (H1)-(H5) are the same for all users, we get similar expressions for the feasible bit-rates of the other users and this collection of bit-rates of the different users is feasible.
Remark 1 Continuous-time. Consider a continuous-time model of the channel (2.1). Let w be the bandwidth of the considered sub-carrier. The results in the discrete-time extend to the continuous-time case, but the capacity bounds, such as the right-hand side of (2.4), should be multiplied by the bandwidth w of the considered sub-carrier.
2.3.3 Ergodic capacity
Consider now a given sub-carrier and multiple time-slots. Recall that we assumed that the fading matrices are ergodic across different time-slots. Then, by the ergodic theorem, the capacity averaged over a large number of time-slots approaches the ergodic capacity E [C] where the expectation is taken with respect to the fading distribution.
1which is consistent with [22, Equation (2)]
2.3. OFDM CELLULAR NETWORK WITH MIMO 21 Corollary 1 The ergodic capacity of the OFDM channel with MIMO (2.1) is lower bounded by
E [C] ≥ E
log
2det
I
rA+ P t
AH
uH
u∗L
u∆
−1(2.5) where ∆ is given by (2.3).
Proof. The result follows by taking the expectation of Equation (2.4) with respect to the fading.
Asymptotic bound
The right-hand side of the Equation (2.5) may be approximated using the following asymptotic result when the number of transmitting and receiving antennas goes to infinity. As will be shown in Appendix 2.1.3, the approximation remains reasonable even for a moderate number of antennas.
Lemma 1 [69, Appendix] Assume that the fading matrix of each base station has i.i.d. centred components with variance 1. (Recall that we have already assumed that H
vare independent across v.) Assume that t
A, r
A→ ∞ such that
rtAA
→ Q, then
1
r
Alog det
I
rA+ P t
AH
uH
u∗L
u
N I
rA+ P t
AX
v6=u
H
vH
v∗L
v
−1
(2.6)
converges almost surely to
Q X
v6=u
log L
v+
NP ηQ1L
v+
NP ηQ2!
+ Q log
1 + P N L
uη
1Q
+ log η
2η
1+ η
1− η
2(2.7)
where η
1and η
2are respectively solutions of
η
1+ X
v
P η
1 PQ
η
1+ N L
v= 1 η
2+ X
v6=u
P η
2 PQ
η
2+ N L
v= 1
The above expressions involve solutions of two non linear equations, which require the knowl-
edge of the received powers from all interfering base stations. In what follows we will establish
another lower bound for the capacity, whose evaluation is much simpler, as simple as the evalu-
ation of the capacity of the AWGN channel, and requires only the knowledge of the interference
power aggregated over all the interfering base stations. We shall compare the two bounds nu-
merically in Section 2.3.3 below.
Jensen’s bound
The following proposition gives a lower bound for the ergodic capacity under the assumption that the covariance of the fading matrix H
vequals identity; that is
E [H
vH
v∗] = I
rA, for all BS v
which means in particular that the fadings of two different transmitting antennas are uncorre- lated.
Proposition 2 Assume that E [H
vH
v∗] = I
rA, for all base station v, then the ergodic capacity E [C] of the channel (2.1) is lower bounded by
E [C] ≥ E [log
2det (I
rA+ SINRH
uH
u∗)] (2.8) where
SINR = (P/t
A) /L
uN + (P/t
A) P
v6=u
1/L
v(2.9) The SINR in the above equation can be seen as the Signal to Interference and Noise Ratio per transmitting antenna.
Proof. Let E [·|H
u] designate the expectation conditionally to H
u. By the properties of the conditional expectation we have
E [C] = E [E [C|H
u]]
Equation (2.4) implies that
E [C|H
u] ≥ E
log
2det
I
rA+ P t
AH
uH
u∗L
u∆
−1H
uwhere ∆ is given by (2.3).
Using Jensen’s inequality and convexity of the function A 7→ log
2det
I
rA+
tPA
HH
∗A
−1on the set of positive definite matrices of C
rA×rA(cf. [40, Lemma II.3]), we deduce that
E [C|H
u] ≥ E [ log
2det (I
rA+ SINRH
uH
u∗)| H
u] where the SINR is given by (2.9). Thus
E [C] = E [E [C|H
u]]
≥ E [E [log
2(1 + SINRH
uH
u∗) |H
u]]
= E [log
2(1 + SINRH
uH
u∗)]
Remark 2 Note that the right-hand side of (2.8) represents the capacity of a MIMO channel with AWGN channel and i.i.d. circularly symmetric Gaussian fading given in Telatar [91, Theorem 1].
Thus, it may be calculated using the analytic formula given in [91, Theorem 2] or approximated with the help of the asymptotic result of Lemma 4 stated in the Appendix as follows
E [C] ≥ E [log
2det (I
rA+ SINRH
uH
u∗)]
' r
Alog (2) C
t
A× SINR, t
Ar
A(2.10)
where C is given by (2.30).
2.4. MMSE 23 Remark 3 Time/frequency diversity. Averaging over a large number of time-slots corresponds to exploiting the so-called time-diversity, which is suitable for the analysis of the performance of a variable bit-rate traffic as observed in [30, §I]. Consider now a given time-slot and large number n of sub-carriers. Assume that the fadings for different sub-carriers are i.i.d., then again, by the law of large numbers, the capacity of a large number n of sub-carriers approaches the ergodic capacity. Thus the ergodic capacity is also appropriate for a constant bit-rate traffic provided the number of sub-carriers allocated to each user is large enough. If the number of sub- carriers allocated to each user is not sufficiently large, then, as observed in [30, §I], a relevant performance indicator is the outage probability, defined as the probability that the capacity in a given time-slot is smaller than the desired bit-rate r. Evaluation of this latter characteristic is not in the scope of this thesis.
Comparison of the lower bounds
We aim now to compare numerically the bounds (2.7) and (2.8). In this regard, we consider a hexagonal cell surrounded by 6 neighboring base stations. The distance between two base stations is 0.5km and the distance propagation law, i.e. path-loss is l (r) = (Kr)
βwhere K = 7764, β = 3.52 which are the typical values in urban areas. We consider that a noise power equals N = −93dBm, standard deviation of shadowing of 8dB and a transmission power of the base station P = 58.5dBm. We consider 2 receiving antennas and a number of transmitting antennas t
A∈ {1, 2, 8}. Figure 2.1 gives the capacity lower bounds (2.7) and (2.8) called, respectively, asymptotic and Jensen bound, as function of the distance between the user and the central base station. This figure shows that the two bounds are close to each other.
0 2 4 6 8 10 12
0 0.05 0.1 0.15 0.2 0.25 0.3
Capacity per receiving ant. [nats/s/Hz]
Distance [km]
Asymptotic bound, 1x2 Jensen bound Asymptotic bound, 2x2 Jensen bound Asymptotic bound, 4x2 Jensen bound Asymptotic bound, 8x2 Jensen bound
Figure 2.1: Jensen (2.8) and asymptotic (2.7) lower bounds for the capacity (peak bit-rate) of the MIMO flat fading channel with additive noise in the downlink of an OFDMA cellular network.
Capacity as function of the distance between the user and his serving base station.
2.4 MMSE
The linear MMSE (Minimum Mean Square Error) decoder of the channel (2.1) means that the receiver estimates the transmitted signal Ψ
nusing a linear transformation Ψ
nof the received signal Y
nminimizing the error E
Ψ
n− Ψ ˆ
n2
. MMSE-SIC (Successive Interference Cancella-
tion) consists in decoding successively the t
Atransmitting antennas while suppressing recursively the previously decoded signals. The objective of the present section is to establish lower bounds for the capacity of MMSE and MMSE-SIC for a cellular network where interference is neither white nor Gaussian.
2.4.1 MMSE capacity given fading
We consider a single time-slot and sub-carrier in this section; thus the fading is assumed given.
It follows from the general theory of linear estimation [28, §3.3] that Ψ ˆ
n= Γ
ΨnYnΓ
−1Yn
Y
n, n ∈ N (2.11)
where Γ
Yn= E [Y
nY
n∗] = Γ
Σ+
tPALu
HH
∗is the covariance matrix of Y
nand Γ
ΨnYn= E [Ψ
nY
n∗] =
P
tAL1/2u
H
∗is the covariance matrix of Ψ
nand Y
n. Equation (2.11) is in fact a system of t
Aequations corresponding to the estimation of the signals emitted by the different transmitting antennas of the serving BS. More specifically, denoting by Ψ
n(k) the signal emitted by the k-th antenna and ˆ Ψ
n(k) the corresponding estimation, Equation (2.11) decomposes into
Ψ ˆ
n(k) = Γ
Ψn(k)YnΓ
−1Yn
Y
n, n ∈ N
where Γ
Ψn(k)Yn= E [Ψ
n(k) Y
n∗]. The above equation may be written in the form Ψ ˆ
n(k) = α
∗Ψ
n(k) + z
nwhere α ∈ C and z
nis a random variable with values in C . The above expression may be seen as the input-output relation of an additive channel corresponding to the k-th transmitting antenna.
It is shown in [92, Equation (8.67)] that the corresponding signal to noise power ratio equals
SNR
k= E h
|α
∗Ψ
n(k)|
2i E h
|z
n|
2i = P
t
AL
uh
∗kΓ
−1kh
k, k = 1, . . . , t
A(2.12) where h
kis the k-th column of the fading matrix H
uand
Γ
k= ∆ +
tA
X
i=1,i6=k
P t
AL
uh
ih
∗i, k = 1, . . . , t
Awhere ∆ is given by (2.3). It follows from Corollary 2 that the capacity of the k-th transmitting antenna (when considering interference from other antennas as well as from other BS) is lower bounded by
C
k≥ log
2(1 + SNR
k) , k = 1, . . . , t
AThus the capacity of the channel is lower bounded by
C
MMSE=
tA
X
k=1
C
k≥
tA
X
k=1
log
2(1 + SNR
k) (2.13)
2.4. MMSE 25
2.4.2 MMSE ergodic capacity
Consider now a given sub-carrier and multiple time-slots. The capacity of the channel is then the expectation of the above capacity (2.13) with respect to fading; thus
E [C
MMSE] ≥
tA
X
k=1
E [log
2(1 + SNR
k)]
=
tA
X
k=1
E [E [ log
2(1 + SNR
k)| H
u]]
Using [40, Lemma II.3] and Jensen’s inequality, it follows that E [C
MMSE] ≥
tA
X
k=1
E
log
21 + P t
AL
uh
∗kΓ ¯
−1kh
k(2.14) where
Γ ¯
k= E [Γ
k|H
u] =
N + P t
AX
v6=u
1 L
v
I
rA+
tA
X
i=1,i6=k
P t
AL
uh
ih
∗iThe right-hand side of (2.14) may be evaluated numerically using Monte Carlo method based on samples of the fading matrix H
u.
2.4.3 MMSE-SIC
As we said previously, MMSE-SIC consists in decoding successively the t
Atransmitting antennas, but before decoding the signal from a given antenna we suppress the previously decoded signals.
Thus the channel for the k-th transmitting antenna is an additive channel with SNR given by (2.12) where the matrix Γ
kis now given by
Γ
k= ∆ +
tA
X
i=k+1
P t
AL
uh
ih
∗i, k = 1, . . . , t
AThe lower bound (2.13) of capacity given the fading remains valid with the above modification of SNR. The lower bound (2.14) of the ergodic capacity holds also true with
Γ ¯
k=
N + P t
AX
v6=u
1 L
v
I
rA+
tA
X
i=k+1
P t
AL
uh
ih
∗iThe proof is based on Jensen inequality and follows the same lines as for MMSE. It follows from [92, Equation (8.71)] that
tA
X
k=1
E
log
21 + P
t
AL
uh
∗kΓ ¯
−1kh
k= E [log
2det (I
rA+ SINRH
uH
u∗)]
where SINR is given by (2.9). Note that the right-hand sides of the above equation and Equa-
tion (2.8) are equal; that is we retrieve the same capacity lower bound as for the original chan-
nel (2.1).
2.5 Numerical results for the link capacity
The objective of the present section is to compare the theoretical expressions established in the previous section to real field measurements and to some simulation compliant with the 3GPP recommendation [3].
2.5.1 Link layer model calibration
We consider firstly a user served by a base station through an additive white Gaussian noise (AWGN) SISO channel neglecting fading and interference for the moment. The user gets ideally (i.e. in the asymptotic sense of information theory) a bit-rate given by the famous Shannon’s formula w log
2(1 + SNR) where w is the bandwidth allocated to the considered user and SNR is the signal to noise power ratio. In order to get rid of the dependence of the bit-rate on the bandwidth, we define the spectral efficiency as the ratio of the bit-rate to the bandwidth which equals log
2(1 + SNR) in the AWGN context.
Mogensen et al. [73], [53] and the 3GPP [4, §A.2] have observed that the LTE system spectral efficiency in this AWGN context is well approximated by
C ' c log
2(1 + qSNR) (2.15)
for some constant c < 1 and q accounting on the one hand for the gap between the practical coding schemes and the optimal ones and on the other hand for the loss of capacity due to signalling. This observation shall be confirmed and the typical value of c and q for LTE will be given.
First, we will calibrate these parameters c and q for real coding schemes considering the simplest AWGN SISO channel, and then use them in the analysis of the MIMO channel with fading and interference.
Note that the relative difference 1 − c for q = 1 between the Shannon’s limit and the practical LTE system may be seen as a progress margin for potential evolution of the technology in the AWGN context.
According to [2, §6.8], [35, p.155] LTE signalling consumes about 30% of the available capacity.
On the other hand, different M -QAM modulations with M ∈ {4, 16, 64} are used with link adaptation and a target block error rate 10
−2. Moreover, CRC and turbo coding are implemented.
The 3GPP [4, §A.2] shows that the bit-rate of LTE is about 25% smaller than the Shannon capacity. In order to account for these losses, we assume
c = (1 − 0.3) × (1 − 0.25) ' 0.5, and q = 1 (2.16) in Equation (2.15).
Figure 2.2 shows that the analytic approximation (2.15) of the SISO capacity with the values of c and q proposed in (2.16) fits well the results of Orange’s link simulation tool in AWGN.
Thus we will retain c = 0.5 and q = 1 to weigh, respectively, the capacity and the SINR in the subsequent analysis of the MIMO channel with fading and interference.
More specifically, using the modified AWGN formula (2.15), the capacity lower bound (2.10) becomes
E [C] ≥ c r
Alog (2) C
t
A× SINR, t
Ar
A(2.17)
where the parameter c is given by (2.16) and the function C is given by Equation (2.30).
2.5. NUMERICAL RESULTS FOR THE LINK CAPACITY 27
0 0.5 1 1.5 2 2.5 3 3.5
-5 0 5 10 15 20
Capacity per unit bandwidth [bit/s/Hz]
SNR [dB]
Simulations Analytic: a=0.5, q=1
